Question

the volume of a rectangular box is $2x(2x + 4)(2x - 2)$. (drawing is not to scale.) image of a rectangular box with labels: length $2x + 4$, width $2x$, height $2x - 2$ which statement about the volume of the box is true? \\(\bigcirc\\) a. the volume is the product of the area of the base, $2x(2x + 4)$, and the height, $2x - 2$. \\(\bigcirc\\) b. the volume is the product of the length, $2x + 4$, and the width, $2x$. \\(\bigcirc\\) c. the volume does not depend on the height, $2x - 2$. \\(\bigcirc\\) d. the volume is the sum of the length, $2x + 4$, the width, $2x$, and the height, $2x - 2$.

Explanation:

To determine the correct statement about the volume of a rectangular box, we use the formula for the volume of a rectangular prism, which is \( V = \text{length} \times \text{width} \times \text{height} \).

• For option A: The area of the base of a rectangular box is \( \text{length} \times \text{width} \), which is \( (2x + 4)(2x) = 2x(2x + 4) \). Then, the volume is the product of the base area and the height (\( 2x - 2 \)), so \( V = 2x(2x + 4)(2x - 2) \), which matches the given volume formula.
• For option B: The product of length and width is the area of the base, not the volume. The volume requires multiplying by the height as well, so this is incorrect.
• For option C: The volume of a rectangular box depends on all three dimensions (length, width, height), so it does depend on the height \( 2x - 2 \). This is incorrect.
• For option D: The volume of a rectangular box is the product of length, width, and height, not the sum. So this is incorrect.

Answer:

A. The volume is the product of the area of the base, \( 2x(2x + 4) \), and the height, \( 2x - 2 \).